Compute the following block products: A11 B11 || 1 0 0 --**-**4] 0 1 -5 5 0 0 -1 -5 -6 Let C = AB. Then C can be computed by block multiplication: c = [C₁1 C1₂] - C = [ C22 1 c=[CH CH₂2]- C21 A = [ALI B = A11 A12 A12B21 B11 B12 B21 B22 -1 = Use them to find C₁1, and then evaluate the remaining blocks of C similarly: 6 -2 -2 4 2 6 3 4 −1 1 1 [A11B11+A12B21 A21B11 + A22B21 -5 5 -5 non 6 5 5 1 -4 0 5 A11B12 + A12B22 A21 B12 + A22 B22
Compute the following block products: A11 B11 || 1 0 0 --**-**4] 0 1 -5 5 0 0 -1 -5 -6 Let C = AB. Then C can be computed by block multiplication: c = [C₁1 C1₂] - C = [ C22 1 c=[CH CH₂2]- C21 A = [ALI B = A11 A12 A12B21 B11 B12 B21 B22 -1 = Use them to find C₁1, and then evaluate the remaining blocks of C similarly: 6 -2 -2 4 2 6 3 4 −1 1 1 [A11B11+A12B21 A21B11 + A22B21 -5 5 -5 non 6 5 5 1 -4 0 5 A11B12 + A12B22 A21 B12 + A22 B22
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Matrix Multiplication Using Block Matrices**
We have two matrices, \( A \) and \( B \), expressed as block matrices:
\[
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 4 & 5 \\ 0 & 1 & -5 & 5 & -4 \\ 0 & 0 & -1 & -5 & -6 \end{bmatrix}
\]
\[
B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} 6 & -2 & -2 & -5 \\ 4 & 2 & 5 & -5 \\ 6 & 3 & 6 & 5 \\ 4 & -1 & 5 & 1 \\ 1 & 1 & -4 & 0 \end{bmatrix}
\]
**Objective:**
Let \( C = AB \). Then \( C \) can be computed by block multiplication:
\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{bmatrix}
\]
**Instructions for Calculation:**
1. **Compute the Block Products:**
- \( A_{11}B_{11} \) (top left matrix block)
- \( A_{12}B_{21} \) (top left supplementary matrix block)
2. **Use these products to find \( C_{11} \), and then evaluate the remaining blocks of \( C \) similarly:**
\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix}
\]
**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2F9311a1b0-21eb-4bc6-9697-6c021a08719a%2Fnz07od2_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Multiplication Using Block Matrices**
We have two matrices, \( A \) and \( B \), expressed as block matrices:
\[
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 4 & 5 \\ 0 & 1 & -5 & 5 & -4 \\ 0 & 0 & -1 & -5 & -6 \end{bmatrix}
\]
\[
B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} 6 & -2 & -2 & -5 \\ 4 & 2 & 5 & -5 \\ 6 & 3 & 6 & 5 \\ 4 & -1 & 5 & 1 \\ 1 & 1 & -4 & 0 \end{bmatrix}
\]
**Objective:**
Let \( C = AB \). Then \( C \) can be computed by block multiplication:
\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{bmatrix}
\]
**Instructions for Calculation:**
1. **Compute the Block Products:**
- \( A_{11}B_{11} \) (top left matrix block)
- \( A_{12}B_{21} \) (top left supplementary matrix block)
2. **Use these products to find \( C_{11} \), and then evaluate the remaining blocks of \( C \) similarly:**
\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix}
\]
**
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